Some Simple Linear Groups

Abstract

The Jordan-Hölder theorem tells us that once we know extensions and simple groups, then we know all finite groups. There arc several infinite families of finite simple groups (in addition to the cyclic groups of prime order and the large alternating groups), and our main concern in this chapter is the most “obvious” of these, the projective unimodular groups, which arise naturally from the group of all matrices of determinant 1 over a field K. Since these groups are finite only when the field K is finite, let us begin by examining the finite fields.KeywordsBilinear FormSimple GroupFinite FieldOrthogonal GroupHyperbolic PlaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.